Splitting subspaces, Krylov subspaces and polynomial matrices over finite fields

Abstract

Let V be a vector space of dimension n over the finite field Fq and T be a linear operator on V . Given an integer m that divides n, an m-dimensional subspace W of V is T-splitting if V = W ⊕ TW ⊕ · · · ⊕ Td−1W where d = n/m. Let σ(m, d; T) denote the number of m-dimensional T-splitting subspaces. Determining σ(m, d; T) for an arbitrary operator T is an interesting problem. We prove that σ(m, d; T) depends only on the similarity class type of T and give an explicit formula in the special case where T is cyclic and nilpotent. Denote by σ(m, d; τ ) the number of m-dimensional splitting subspaces for a linear operator of similarity class type τ over an Fq-vector space of dimension md. For fixed values of m, d and τ , we show that σ(m, d; τ ) is a polynomial in q. This problem is closely related to another open problem on Krylov spaces. We discuss this connection and give explicit formulae for σ(m, d; T) in the case where the invariant factors of T satisfy certain degree conditions. A connection with another enumeration problem on polynomial matrices is also discussed. We finally present a brief review of some recent developments in the splitting subspaces problem. In particular, the connection of this problem to the theory of symmetric functions is highlighted. Lastly, we conclude with some future directions for research.